This document discusses the theoretical calculation of ionization cross sections for the K-shell, L-shell, and M-shell of bismuth atoms due to electron impacts. The Khare model was modified to calculate these cross sections from the ionization threshold to 1 GeV. The calculated cross sections show good agreement with available experimental data for K-shell and L-shell ionization but overestimate experimental data for M-shell ionization. The modifications improve agreement between theory and experiment over a wide energy range.

1. K-Shell L-Shell and M-Shell Ionization Cross Sections of Bismuth Atom by Electron Impacts, Manoj
Kumar, Journal Impact Factor (2015): 8.5041 (Calculated by GISI) www.jifactor.com
www.iaeme.com/ijaret.asp 34 editor@iaeme.com
1
Department of physics, Meerut College, Meerut –250001, INDIA
ABSTRACT
The theoretical model, developed by Khare model, has been modified to calculate the total
cross sections for K-shell L-shell and M-shell ionization of Bismuth (Z=83) atom due to electron
impact at incident electron energy from ionization threshold to 1 GeV. The various calculated cross
sections are in remarkable agreement with available experimental data and other theoretical cross
sections.
PACS No: 34.80Dp
Key word: Atomic Ionization, Electron Impact.
INTRODUCTION
Electron impact ionization cross sections for inner shells are needed in many areas of
research such as fusion physics, plasma physics, space physics environmental protection, material
analysis by various techniques etc. [1-2]. The ionization cross sections find important application in
field such as atmosphere physics, radiation science, astrophysics, etc.
Past six decades many experimental and theoretical studies have been carried out to estimate
the electron impact inner-shell ionization cross sections by various groups. Experimentally inner
shell ionization cross sections have been measured by Ishii et al [3], scholz et al [4], Hoffmann et al
[5], Middleman et al [6], Palinkas et al [7], etc. at different times by electron impact.
Many theoretical researchers like Casnati et al. [8], Hombourger et al. [9], Scofield [10],
Khare et al [11-13], Kim et al [14], Haque et al [15], etc. have calculated inner ionization cross
sections due to electron impact. Scofield [10] proposed a model to calculate the ionization cross
sections over wide incident energies taking into account the relativistic effect in relativistic plane
wave born approximation (PWBA) through Dirac equation. His cross sections exhibit nice
agreement with experimental data at ultrarelativistic energies. However, these methods fail at impact
energies near threshold of ionization. Khare et al. [11-13] have calculated the electron impact
ionization cross sections for Inner-shell for a numbers of atoms. They have employed the Plane
Wave Born Approximation (PWBA) with corrections for exchange, coulomb and relativistic effects.
Along with the longitudinal interaction, the contribution of transverse interaction to the ionization
cross section is also included. Recently many researchers like Haque et al. [15] etc. have calculated
K-SHELL L-SHELL AND M-SHELL IONIZATION CROSS
SECTIONS OF BISMUTH ATOM BY ELECTRON IMPACTS
Manoj kumar1
Volume 6, Issue 6, June (2015), pp. 34-39
Article ID: 20120150606006
International Journal of Advanced Research in Engineering and
Technology (IJARET)
© IAEME: www.iaeme.com/ijaret.asp
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
IJARET
© I A E M E

2. K-Shell L-Shell and M-Shell Ionization Cross Sections of Bismuth Atom by Electron Impacts, Manoj
Kumar, Journal Impact Factor (2015): 8.5041 (Calculated by GISI) www.jifactor.com
www.iaeme.com/ijaret.asp 35 editor@iaeme.com
the inner shell ionization cross sections by modifying the different model from threshold to ultra
relativistic range.
In 1999 Khare et al [17] purposed model to calculate the ionization cross sections for
molecules by combining the useful features of two models Kim et al [18] and Saksena et al [19]. In
Khare (BEB) method, calculated cross sections [20-22] were in between in better agreement with the
experiment data over a wide energy varying from threshold to several MeV. This model have been
modified by Y. Kumar et al [23-24] to obtained the K shell cross sections for many atoms (6<Z<92)
in the energy varying from the threshold of ionization to 1 GeV by electron impact. They have
replaced factor 1/(Er+I+U) by 1/(Er +f), where Er = relativistic energy of the incident particle, I =
target particle’s binding energy, U = target particle’s kinetic energy and f = ηIr/(1+ξZ) is a factor
which depends on atomic number (Z), here η and ξ are the experimentally fitted parameter and Ir =
target particle’s binding energy with relativistic correction. The calculated cross sections were in
good agreement with available experimental data.
In present investigation we have considered that factor f depends angular quantum number (l)
and replaced f by f/(l+1) for l sub-shell ionization in modified Khare (BEB) model [23] to calculate
L shell ionization cross sections of gold atom due to the electron and positron impact.
Theory
In the modified Khare [BEB] model [24] for Li subshell the total ionization cross section is
given by
iT iPM iPB itσ σ σ σ= + + (1)
With the Mott cross section
( ) ( )
2
22 2 2
2 1 5 1 1 1 1
1 ln
1 2 1 21
iPM
s t t t t t
t f t t t t t tt
σ
   − − + + + 
= − + + − − −    
+ + +  +    (2)
Bethe cross section
2 2 2 2
1 1 1 1
4431 1 ln ln 1
2 2 2 2
r r
iPB
I Es
t f t t mc t mc
σ
      
= ⋅ − − + + +        +        
(3)
and the cross section due to transverse interaction is
( )2 2
ln 1nl
it
sb
t f
σ β β
 
 = − − +   + 
(4)
Where r
r
E
t
I
= ,
Relativistic energy of projectile of mass m, 2
2
1 1
1
2
1
rE mc
E
mc
 
 
 = −
  +    

3. K-Shell L-Shell and M-Shell Ionization Cross Sections of Bismuth Atom by Electron Impacts, Manoj
Kumar, Journal Impact Factor (2015): 8.5041 (Calculated by GISI) www.jifactor.com
www.iaeme.com/ijaret.asp 36 editor@iaeme.com
ionization energy with relativistic correction , 2
2
1 1
1
2
1
2
r
nl
I mc
I
mc
 
 
 = −
  +    
and f = ηIr/{(l+1)(1+ξZ)}, here η=1.77[1+(q/Z)2
], q = number of electron filled up to nlth
sub
shell and ξ =.018 are the experimentally fitted parameter [26].
In this paper we have c=velocity of light, E=Projectile energy, m=rest mass of electron,
nlb =Bethe parameter of nl subshell,
2 2
0
2
4 i
r
a R N
S
I
π
= , R=Rydberg energy, iN =number of electrons in
th
i subshell, 0a =first Bohr radius, nlI =ionization thresholds energy of nl subshell. l= angular
quantum number, β = v/c, v=velocity of the incident particle. The value of bnl in the Khare
parameter[5] is given by
bnl=αp-γ
(5)
Where p=I/Is, Is=Zs
2
R/n2
Zs =Z-s is the effective atomic number and s is the screening parameter
Bethe parameters nlb [25-26] are given in table.
Parameters/Subshells K L1 L2 L3 M1 M2 M3 M4 M5
α .285 .220 .158 .153 .180 .184 .189 .0915 .0935
γ 1.7 1 1.8 1.54 .918 1.10 1.08 3.66 3.63
RESULT AND DISCUSSION
In present investigation the K-shell, L-shell and M-shell ionization cross sections have been
calculated for Bi (83) atom by modified the Khare [BEB] model. The ionization potentials are taken
from Deslaux [27].
Figure1 to 3 show the inner shell ionization cross sections obtained in modified Khare model.
The experimental data for electron impact are shown in figure comparison.
In fig.1 the K-shell ionization cross sections for Bi have been shown. The obtained results are
in contrast with measured values of Hoffmann et al. [5], Ishii et al. [3] and Scholz et al. [4], while
data by Middleman et al. [6] is slightly higher than the calculated values. Theoretical results of
Hombourger et al. [9] and Casnati et al. [8] do not agree with experimental data of Hoffmann et al.
[5], Ishii et al. [3] and Middleman et al. [6] at high energies.
In fig.2 we have compared the data of Bi for L-shell. There are four experimental data, named
Palinka et al [7], Hoffman et al [5], Ishii et al [3] and Middleman et al [6]. The present results are
good agreement with data measured by Palinka et al [7], and Ishii et al [3] within 7% over entire
energy range. While the present values are slightly lower than the data by Hoffmann et al [5]. The
calculated cross sections agree with the experimental results of Middleman et al [6] at high energies.
The figure 3 shows the present ionization cross sections and experimental data measured by
Hoffmann et al. [5] for M-shell ionization. The figure does not show good agreement calculated
cross sections and experimental data.
In the total ionization cross sections, the transverse interaction given by equation (4) between
the projectile and target is included and it is fond to be of great significance for E larger than 1MeV.
For K and L shells good agreement between the theory and various experimental results for electron

4. K-Shell L-Shell and M-Shell Ionization Cross Sections of Bismuth Atom by Electron Impacts, Manoj
Kumar, Journal Impact Factor (2015): 8.5041 (Calculated by GISI) www.jifactor.com
www.iaeme.com/ijaret.asp 37 editor@iaeme.com
impact ionization cross sections has been obtained for U atom over the whole range. However, for M
shell the present theoretical approach overestimates the experimental results by factor of 4. This
disagreement is due to Bethe parameter calculated by Khare et al. [25-26]. They have taken the
hydrogenic generalized oscillator strength to obtained the Bethe parameter bnl . With the continuum
optical oscillator strengths df(w,o)/dw=NI/w2
,where w is the energy loss, the value of the Bethe
parameter bnl is equal to .5. With this value of bnl, the calculated cross sections are also shown in the
figure, which shows the excellent agreement with the experimental data of Hoffmann et al. [5].
0
5
10
15
20
25
30
35
10 100 1000 10000 100000 1000000
E(keV)
Present
Hombourger
Casnati
Scholz
Hofmann
Middleman
Ishii
Fig: 1
Figure 1 The present theoretical electron impact ionization cross section of Bi for K,shell depicts by
line(━) compared with the experimental results given by Hoffmann et al.(◾),Ishii et al.(♦) and
Scholz et al.(▴) and theoretical results calculated by Casnati et al.(….), Hombourger et al.(---) and
Middleman et al. depicts by (▫)
0
2
4
6
8
10
12
14
16
10 100 1000 10000 100000 1000000
E(keV)
Present
Palinkas et al
Ishii et al
Hoffmann et
al
Fig: 2
Figure 2 The present theoretical electron impact ionization cross section of Bi for K,shell depicts by
line(━) compared with the experimential results given by Hoffmann et al.(▴),Ishii et al.(▪) and
Palinkas et al.(♦)
σ(10-28
m2
)σ(10-26
m2
)

5. K-Shell L-Shell and M-Shell Ionization Cross Sections of Bismuth Atom by Electron Impacts, Manoj
Kumar, Journal Impact Factor (2015): 8.5041 (Calculated by GISI) www.jifactor.com
www.iaeme.com/ijaret.asp 38 editor@iaeme.com
0
2
4
6
8
10
1 10 100 1000 10000 100000 1E+06 1E+07
E(KeV)
Present
Hoffmann
Present(b=.5)
Ishii
Fig: 3
The figure 3 shows the present theoretical ionization cross section of Bi for M-shell depicts by (—)
,when(b=.5) depict,(…..), compared with the Hoffmann et.al result given by Hoffmann shown(▪)
and Ishii el.at(▴)
CONCLUSION
Finally, we conclude that slight modifications in Khare (BEB) model has considerable
improved the agreement between the theory and the experiment data. The available experimental
data are in satisfactory agreement with the present cross section over a wide range. The application
of the present model to the ionization of inner shell for other atoms is of interest.
ACKNOWLEDGMENT
The authors are grateful to Ex. Prof. S. P. Khare of C. C. S. University, Meerut, India for
fruitful discussion.
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www.iaeme.com/ijaret.asp 39 editor@iaeme.com
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